(ignored inputs)COMMENT Cops #647 - #721: generated ground TRSs; evenly distributed in the UNR/UNC/NFP/CR hierarchy submitted by: Bertram Felgenhauer
Rewrite Rules:
   [ b -> h(c,f(b)),
     a -> c,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),b),b))))) -> a,
     c -> h(f(h(c,f(f(c)))),f(b)) ]
Apply Direct Methods...
Inner CPs:
   [ h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(b),b),b))))) = a ]
Outer CPs:
   [  ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
inner CP cond (upside-parallel)
innter CP Cond (outside)
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ a = h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),b),b))))),
     a = h(h(h(f(a),a),a),f(f(f(h(h(f(b),h(c,f(b))),b))))),
     a = h(h(h(f(a),a),a),f(f(f(h(h(f(b),b),h(c,f(b))))))),
     a = h(h(h(f(c),a),a),f(f(f(h(h(f(b),b),b))))),
     a = h(h(h(f(a),c),a),f(f(f(h(h(f(b),b),b))))),
     a = h(h(h(f(a),a),c),f(f(f(h(h(f(b),b),b))))),
     h(h(h(f(c),c),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),c),c),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),c),a),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(c),a),c),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(a),c),c),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),h(c,f(b))),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),b),h(c,f(b))))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),h(c,f(b))),h(c,f(b))))))) = a,
     h(h(h(f(c),a),a),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),c),a),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),a),c),f(f(f(h(h(f(b),b),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),b),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),h(c,f(b))),b))))) = a,
     h(h(h(f(a),a),a),f(f(f(h(h(f(b),b),h(c,f(b))))))) = a ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <h(h(h(f(a),a),a),f(f(f(h(h(f(h(c,f(b))),b),b))))), a> by Rules <0, 2> preceded by [(h,2),(f,1),(f,1),(f,1),(h,1),(h,1),(f,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 11 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
<F,Q,Q_f,Delta> where
   F = {(a,0),(b,0),(c,0),(f,1),(h,2)}
   Q = {q_0,q_1,q_{f(h(c,f(f(c))))},q_{f(f(h(h(f(b),b),b)))},q_{a},q_{b},q_{c},q_{f(a)},q_{f(b)},q_{f(c)},q_{f(f(c))},q_{f(f(f(h(h(f(b),b),b))))},q_{h(h(f(b),b),b)},q_{f(h(h(f(b),b),b))},q_{h(h(f(a),a),c)},q_{h(f(a),a)},q_{h(f(b),b)},q_{h(c,f(f(c)))}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_1,
     a -> q_{a},
     b -> q_{b},
     c -> q_1,
     c -> q_{a},
     c -> q_{c},
     f(q_{f(f(h(h(f(b),b),b)))}) -> q_{f(f(f(h(h(f(b),b),b))))},
     f(q_{a}) -> q_{f(a)},
     f(q_{b}) -> q_{f(b)},
     f(q_{c}) -> q_{f(c)},
     f(q_{f(c)}) -> q_{f(f(c))},
     f(q_{h(h(f(b),b),b)}) -> q_{f(h(h(f(b),b),b))},
     f(q_{f(h(h(f(b),b),b))}) -> q_{f(f(h(h(f(b),b),b)))},
     f(q_{h(c,f(f(c)))}) -> q_{f(h(c,f(f(c))))},
     h(q_{f(h(c,f(f(c))))},q_{f(b)}) -> q_1,
     h(q_{f(h(c,f(f(c))))},q_{f(b)}) -> q_{a},
     h(q_{f(h(c,f(f(c))))},q_{f(b)}) -> q_{c},
     h(q_{c},q_{f(b)}) -> q_{b},
     h(q_{c},q_{f(f(c))}) -> q_{h(c,f(f(c)))},
     h(q_{f(a)},q_{a}) -> q_{h(f(a),a)},
     h(q_{f(b)},q_{b}) -> q_{h(f(b),b)},
     h(q_{h(h(f(a),a),c)},q_{f(f(f(h(h(f(b),b),b))))}) -> q_0,
     h(q_{h(f(a),a)},q_{c}) -> q_{h(h(f(a),a),c)},
     h(q_{h(f(b),b)},q_{b}) -> q_{h(h(f(b),b),b)} ]
Witness for Non-Confluence: <h(h(h(f(a),a),c),f(f(f(h(h(f(b),b),b))))), a>
Direct Methods: not CR

Combined result: not CR
686.trs: Success(not CR)
NO
(1517 msec.)